SNNAP, the Simulator of Neuronal
Nets and Action Potentials, uses Hodgkin-Huxley type differential
equations to simulate a cell's conductances that in turn may be modulated
by second messengers or ion pools. SNNAP can simulate networks of up
to 100 cells and 300 electrical, chemical and modulatory synaptic. Time
dependent synaptic conductances are described by second-order ordinary
differential equations. The experimenter can then perform virtual elektrophysiological
experiments or observe the net's spontaneous behavior.

I can provide a brief SNNAP
text overview or a colorful poster with the title "Simulating
Physiological and morphological Properties of Neurons with SNNAP (Simulator
for Neural Networks and Action Potentials)". Download the abstract
in HTML or the
poster in PDF.

My first successful simulation was a network of two identical
neurons that exhibit postinhibitory rebound firing, i.e. they fire after
they have been hyperpolarized. To do this I equipped the cells with
a non-deactivating, cation driven inward rectifier, that activates upon
hyperpolarization: the "h current". Both cells are connected
to each other by an inhibitory synapse, i.e. cell A fires and inhibites
(hyperpolarizes) cell B and if cell B fires it inhibits cell A. Obviously,
this leads to rhythmic alternating firing if one of the cells if briefly
hyperpolarized by injecting negative current:

(click on image to enlarge)

Interestingly, it was possible to adjust synaptic strength,
the time constant of the synaptic current and the h-current so that
the number of spikes would be proportional to the intensity (duration
or Amperage) of the stimulating current:

In this run of the simulation, the current was chosen
such that the cells would alternatingly fire single spikes. Of course
these simulations only reflect the behavior of potentially 'real' cells.
None of the values have been adjusted to match any particular biologically
meaningful data. The cells simulated here are purely hypothetical.